Forcing for independence proofs

Dr. Andrew Brooke-Taylor, Andrew.Brooke-Taylor(at)bristol.ac.uk.
Taught Course Centre, Thursdays 15:00-17:00, October - December 2010.

Homework exercises

Lecture notes and whiteboard images

• Lecture 1 notes (typed).
• Lecture 2 whiteboard images
• Lecture 3 notes (scanned from my hand-written notes). Also, the typed up definition of forcing*. I highly recommend proving the final lemma in the notes (the 3 equivalent statements) as an exercise - there are many parts, but it's not all formal manipulations.
• Lecture 4 whiteboard images, Oxford style
• Lecture 5 whiteboard images
• In lecture 6 we talked about nice names. See Kunen.
• Lecture 7 whiteboard images
• Lecture 8 whiteboard images

Summary from before the course

Forcing is the key technique used by set theorists in proving independence proofs, and its development by Paul Cohen won him the Fields Medal in 1966. In this course I will show how it can be used to prove that the Continuum Hypothesis is independent of the standard ZFC axioms for set theory, and to prove that the Axiom of Choice is independent of the other axioms.

No prior knowledge of set theory will be assumed, however a basic level of mathematical logic (eg what a first order formula is) would be helpful.

Outline:

1. Introduction: what independence means (definition, not philosophy!), models of set theory.
2. Generic extensions, the forcing relation
3. Definability of the forcing relation
4. The Truth Lemma, extensions satisfy ZFC
5. Chain conditions, Con(ZFC) implies Con(ZFC + ~CH)
6. Closure conditions, Con(ZFC) implies Con(ZFC + CH)

1. Symmetric models, Con(ZF) implies Con(ZF + ~AC)
2. Looking ahead and at other techniques: ways to prove Con(ZF) implies Con(ZFC)

References